EasyToolTip() {
UIManager.put("ToolTip.background", new ColorUIResource(Color.CYAN));
UIManager.put("TextArea.background", new ColorUIResource(25, 25, 25));
setToolTipText("This is also a way to change the color of the tooltip in your application!");
add(t);
}

The order in which code gets called is what matters here since you are creating your JTextArea before the UIManager has had a chance to change the default background. Change that and your program will change its display:

EasyToolTip() {
UIManager.put("ToolTip.background", new ColorUIResource(Color.CYAN));
UIManager.put("TextArea.background", new ColorUIResource(25, 25, 25));
t = new JTextArea(10, 10);
setToolTipText("This is also a way to change the color of the tooltip in your application!");
add(t);
}